physical realizations of qc @ tehran, jan. 2009 1 mikio nakahara, research centre for quantum...

Physical Realizations of QC @ Tehran, Jan. 2009

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Liquid State NMR Liquid State NMR Quantum Quantum ComputingComputing

Mikio Nakahara,Mikio Nakahara,

Research Centre for Quantum Research Centre for Quantum Computing, Kinki University, JapanComputing, Kinki University, Japan

Financial supports from Kinki Univ.,

MEXT and JSPS

Physical Realizations of QC @ Tehran, Jan. 2009

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Plan of Talk

1. Introduction 2. NMR2. NMR 3. NMR Hamiltonian 4. Gate Operations 5. Pseudopure State 6. Measurement 7. DiVincenzo Criteria 8. Summary

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1. Introduction

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Qubits in NMR Molecule

Trichloroethylene

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Plan of Talk

1. Introduction 2. NMR2. NMR 3. NMR Hamiltonian 4. Gate Operations 5. Pseudo-Pure State 6. Measurement 7. DiVincenzo Criteria 8. Summary

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NMR (Nuclear Magnetic Resonance )　=MRI (Magnetic Resonance Imaging)

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NMR

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Schematic of NMR

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Molecules used in NMR QC

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Plan of Talk

1. Introduction 2. NMR2. NMR 3. NMR Hamiltonian 4. Gate Operations 5. Pseudopure State 6. Measurement 7. DiVincenzo Criteria 8. Summary

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3.1 Single-Qubit Hamiltonian

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Hamiltonian in Rotating Frame

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2-Qubit Hamiltonian

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Plan of Talk

1. Introduction 2. NMR2. NMR 3. NMR Hamiltonian 4. Gate Operations 5. Pseudopure State 6. Measurement 7. DiVincenzo Criteria 8. Summary

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1-Qubit Gates

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Example: Hadamard gate

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Example: Hadamard gate 2

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Selective addressing

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In resonance: .

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0 .1

0 .0

0 .1

0 .1

0 .0

0 .1

0 .9900 .9951 .000

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0 .05

0 .00

0 .05

0 .05

0 .00

0 .05

0 .9980 .9991 .000

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2-Qubit Gates: CNOT

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Plan of Talk

1. Introduction 2. NMR2. NMR 3. NMR Hamiltonian 4. Gate Operations 5. Pseudopure State 6. Measurement 7. DiVincenzo Criteria 8. Summary

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Spins are in mixed state!

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Preparation of a pseudopure state in terms of temporal average method

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Temporal average method

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Averaging three contributions

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Plan of Talk

1. Introduction 2. NMR2. NMR 3. NMR Hamiltonian 4. Gate Operations 5. Pseudopure State 6. Measurement 7. DiVincenzo Criteria 8. Summary

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6.1 Free Induction Decay (FID)

|00〉 |01〉

|10〉 |11〉

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Free Induction Decay (FID)

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6.2 Quantum State Tomography We want to “measure” the density matrix. Measure observable such as

magnetizations to find linear combinations of the matrix elements of the density matrix.

Not enough equations are obtained. Deform the density matrix with pulses to

obtain enough number of equations.

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2-Qubit QST

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Plan of Talk

1. Introduction 2. NMR2. NMR 3. NMR Hamiltonian 4. Gate Operations 5. Pseudopure State 6. Measurement 7. DiVincenzo Criteria 8. Summary

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DiVincenzo Criteria for NMR QCDiVincenzo Criteria for NMR QC A scalable physical system with well charactA scalable physical system with well charact

erized qubits.erized qubits. The ability to initialize the state of the qubits The ability to initialize the state of the qubits

to a simple fiducial state, such as |00…0>.to a simple fiducial state, such as |00…0>. Long decoherence times, much longer than Long decoherence times, much longer than

the gate operation time.the gate operation time. A “universal” set of quantum gates.A “universal” set of quantum gates. A qubit-specific measurement capability.A qubit-specific measurement capability.

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Scalability Selective addressing to each qubit becom

es harder and hader as the # of qubits increases. Limited # of nuclear spices and overlap of resonance freqs.

Signal strength is suppressed as the # of qubits increases. Readout problem.

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Initialization (pseudopure state) # of steps required to prepare a pseudopur

e state increases exponentially as the # of qubits increases.

No real entanglement

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Long decoherence time Decoherence time Single-qubit gate operation time Two-qubit gate op. time May execute Shor’s algorithm for 21=3X7.

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A “universal” set of quantum gates.A “universal” set of quantum gates. One-qubit gates by Rabi oscillation. Two-qubit gates by J-coupling. Cannot turn off interactions; reforcusing te

chnique becomes complicated as the # of qubits increases.

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Measurement capability.Measurement capability.

FID is a well-established techunique. Quantum State Tomograpy and Quantum

Process Tomography are OK. S/N scales as , which limits

the # of qubits to ~ 10.

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Still… NMR QC is commercially available. It can execute small scale quantum algorithms. It serves as a test bed for a real QC to come. May ideas in other realizations are inspired from

NMR. We use NMR QC to demonstrate theoretical

ideas, such as decoherence suppression, optimal control of a Hamiltonian etc.

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Plan of Talk

1. Introduction 2. NMR2. NMR 3. NMR Hamiltonian 4. Gate Operations 5. Pseudopure State 6. Measurement 7. DiVincenzo Criteria 8. Summary

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Liquid state NMR QC is based on a well-established technology. Most of the materials introduced here have been already known in the NMR community for decades.

There are still many papers on NMR QC. It is required to find a breakthrogh for a liquid sta

te NMR to be a candidate of a working QC. ENDOR, Solid state NMR… Thank you for your attention.